Optimal. Leaf size=101 \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+\frac{4}{\sqrt{x^2-2 x-3}+x}+\frac{3}{4 \left (\sqrt{x^2-2 x-3}+x\right )^2}+6 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-6 \log \left (\sqrt{x^2-2 x-3}+x\right ) \]
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Rubi [A] time = 0.0822887, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+\frac{4}{\sqrt{x^2-2 x-3}+x}+\frac{3}{4 \left (\sqrt{x^2-2 x-3}+x\right )^2}+6 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-6 \log \left (\sqrt{x^2-2 x-3}+x\right ) \]
Antiderivative was successfully verified.
[In] Int[(x + Sqrt[-3 - 2*x + x^2])^(-3),x]
[Out]
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Rubi in Sympy [A] time = 5.1796, size = 85, normalized size = 0.84 \[ - 6 \log{\left (x + \sqrt{x^{2} - 2 x - 3} \right )} + 6 \log{\left (- x - \sqrt{x^{2} - 2 x - 3} + 1 \right )} - \frac{2}{- x - \sqrt{x^{2} - 2 x - 3} + 1} + \frac{4}{x + \sqrt{x^{2} - 2 x - 3}} + \frac{3}{4 \left (x + \sqrt{x^{2} - 2 x - 3}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x+(x**2-2*x-3)**(1/2))**3,x)
[Out]
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Mathematica [A] time = 0.0820744, size = 111, normalized size = 1.1 \[ -\frac{\sqrt{x^2-2 x-3} \left (4 x^2+31 x+33\right )}{2 (2 x+3)^2}+3 \log \left (-3 \sqrt{x^2-2 x-3}+5 x+3\right )+3 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )+\frac{x}{2}-\frac{9}{2 x+3}+\frac{27}{8 (2 x+3)^2}-6 \log (2 x+3) \]
Antiderivative was successfully verified.
[In] Integrate[(x + Sqrt[-3 - 2*x + x^2])^(-3),x]
[Out]
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Maple [A] time = 0.033, size = 146, normalized size = 1.5 \[ -9\, \left ( 3+2\,x \right ) ^{-1}-3\,\ln \left ( 3+2\,x \right ) +{\frac{x}{2}}+{\frac{27}{8\, \left ( 3+2\,x \right ) ^{2}}}-{\frac{1}{2} \left ( \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}+3\,{\it Artanh} \left ( 2/3\,{\frac{-3-5\,x}{\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}}} \right ) +{\frac{2\,x-2}{4}\sqrt{ \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}}}}+3\,\ln \left ( -1+x+\sqrt{ \left ( x+3/2 \right ) ^{2}-5\,x-{\frac{21}{4}}} \right ) +{\frac{1}{4} \left ( \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x+(x^2-2*x-3)^(1/2))^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x + \sqrt{x^{2} - 2 \, x - 3}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 - 2*x - 3))^(-3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.26654, size = 431, normalized size = 4.27 \[ \frac{16 \, x^{6} - 4 \, x^{5} - 300 \, x^{4} + 159 \, x^{3} + 931 \, x^{2} - 12 \,{\left (4 \, x^{5} - 27 \, x^{3} - 19 \, x^{2} -{\left (4 \, x^{4} + 4 \, x^{3} - 15 \, x^{2} - 18 \, x\right )} \sqrt{x^{2} - 2 \, x - 3} + 24 \, x + 18\right )} \log \left (x^{2} - \sqrt{x^{2} - 2 \, x - 3}{\left (x + 1\right )} - 3\right ) - 12 \,{\left (4 \, x^{5} - 27 \, x^{3} - 19 \, x^{2} + 24 \, x + 18\right )} \log \left (2 \, x + 3\right ) + 12 \,{\left (4 \, x^{5} - 27 \, x^{3} - 19 \, x^{2} -{\left (4 \, x^{4} + 4 \, x^{3} - 15 \, x^{2} - 18 \, x\right )} \sqrt{x^{2} - 2 \, x - 3} + 24 \, x + 18\right )} \log \left (-x + \sqrt{x^{2} - 2 \, x - 3}\right ) -{\left (16 \, x^{5} + 12 \, x^{4} - 256 \, x^{3} - 41 \, x^{2} - 12 \,{\left (4 \, x^{4} + 4 \, x^{3} - 15 \, x^{2} - 18 \, x\right )} \log \left (2 \, x + 3\right ) + 466 \, x + 132\right )} \sqrt{x^{2} - 2 \, x - 3} + 84 \, x - 342}{4 \,{\left (4 \, x^{5} - 27 \, x^{3} - 19 \, x^{2} -{\left (4 \, x^{4} + 4 \, x^{3} - 15 \, x^{2} - 18 \, x\right )} \sqrt{x^{2} - 2 \, x - 3} + 24 \, x + 18\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 - 2*x - 3))^(-3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x + \sqrt{x^{2} - 2 x - 3}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x+(x**2-2*x-3)**(1/2))**3,x)
[Out]
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GIAC/XCAS [A] time = 0.274239, size = 248, normalized size = 2.46 \[ \frac{1}{2} \, x - \frac{1}{2} \, \sqrt{x^{2} - 2 \, x - 3} - \frac{104 \,{\left (x - \sqrt{x^{2} - 2 \, x - 3}\right )}^{3} + 315 \,{\left (x - \sqrt{x^{2} - 2 \, x - 3}\right )}^{2} + 162 \, x - 162 \, \sqrt{x^{2} - 2 \, x - 3} + 27}{8 \,{\left ({\left (x - \sqrt{x^{2} - 2 \, x - 3}\right )}^{2} + 3 \, x - 3 \, \sqrt{x^{2} - 2 \, x - 3}\right )}^{2}} - \frac{9 \,{\left (16 \, x + 21\right )}}{8 \,{\left (2 \, x + 3\right )}^{2}} - 3 \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - 3 \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} + 1 \right |}\right ) + 3 \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} \right |}\right ) - 3 \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 - 2*x - 3))^(-3),x, algorithm="giac")
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